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How many network motifs can you make with 5 retinal neurons?

Assume that you have 5 kinds of cells: 2 different BCs (Bi, Bj), 2 different GCs (Gi, Gj), 1 AC (A) connecting them.

How many different synaptic motifs can you make?
Answer = 90 as a lumped parameter model
Answer = 2880 as a distributed parameter model

Explanation:
The vertical channels are Bi > Gi; Bj > Gj
The lateral channels are:
A > none: processes 0 and 1
A > Bi,Bj (feedback): processes 2 and 3
A > Gi, Gj (feedforward): processes 4 and 5
A > A (nested feedback): loop x

Start with a summary of "supermotifs"
There are 12 allowed BCAC supermotifs:
motif label
0 i mono input, no feedback
1 j mono input, no feedback
01 dual input, no feedback
02 i mono input, in-channel feedback
03 i mono input, cross-channel feedback
12 j mono input, cross-channel feedback
13 j mono input, in-channel feedback
012 dual input, feedback i
013 dual input, feedback j
023 i mono input, dual feedback
123 j mono input, dual feedback
0123 dual input, dual feedback

There are 4 ACGC feedforward supermotifs:
A > none, A > Gi, A > Gj, A > Gi, Gj

There are 2 ACAC nested feedback supermotifs:
A > none and A > A

The combined motif number is
(((BCAC)(GCAC))-3)(ACAC) = (((12)(4)-3)(2) = 90
(-3 because motifs 0,1, 01 cannot have AC > none)

This presumes that ACAC can be represented as a lumped parameter.

If not, then the site of the ACAC loop matters and we must calculate a
distributed parameter model.

___________________________

The distributed parameter model.
There are 16 nested feedback supermotifs:
0x, 1x, 2x, 3x, 01x, 02x, 03x, 12x, 13x, 23x, 012x, 013x, 023x, 123x, 0123x
There are 4 nested forward supermotifs:
none, 4x, 5x, 45x

Thus there are 64 possible local nested supermotifs.
The total number of motifs becomes 45*64 = 2880

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